Order Accurate Scheme Research Articles

Back and forth error compensation and correction method for linear hyperbolic systems with application to the Maxwell's equations

We study the Back and Forth Error Compensation and Correction (BFECC) method for linear hyperbolic systems and in particular for the Maxwell's equations. BFECC has been applied to schemes for scalar advection equations to improve their stability and order of accuracy. Similar results have been established in this paper for linear hyperbolic systems with constant coefficients. We apply BFECC to the central difference scheme, Lax-Friedrichs scheme and a combination of them for the Maxwell's equations and obtain second order accurate schemes with large CFL numbers (greater than 1 in one or two dimensions). The method is further applied to schemes on non-orthogonal unstructured grids. The new BFECC schemes for the Maxwell's equations operate on a single non-staggered grid and are simple to implement on unstructured grids. Numerical examples are given to demonstrate the effectiveness of the new schemes.

A Stabilized Fourier Spectral Method for the Fractional Cahn-Hilliard Equation

In this paper, the second order accurate (in time) energy stable numerical schemes are presented for the Fractional Cahn-Hilliard (CH) equation. Combining the stabilized technique, we apply the implicit Crank-Nicolson formula (CN) to derive second order temporal accuracy, and we use the Fourier spectral method for space discrete to obtain the fully discretization schemes. It is shown that the schemes are unconditionally energy stable. A few numerical experiments are presented to conclude the article.

Finite element solution of a new formulation for gas flow in a pipe with source terms

A new formulation for isothermal gas flow in a pipeline with gravity and friction source terms is introduced and solved by finite element methods using FEniCS open-source computing platform. The proposed formulation can be easily adapted to simulate different types of real boundary conditions and involves only one scalar unknown instead of two ones (typically mass flux and pressure or density). The main motivation for presenting the work is to find a compact formulation which is able to be solved with a high order accuracy scheme just by changing the type of elements. Also, it can be extended to gas network easily. In order to check the performance of the numerical method, it has been applied to several test problems. At first, with the purpose of finding the order of accuracy, two manufactured tests for different types of boundary conditions are solved. The results show that, for nodal finite elements of degree one, the order of accuracy in space is more than two and for time is between two and three. Then, two test cases with step boundary conditions are solved. The corresponding numerical results show that, even with large time-step and coarse mesh, they are in good agreement with those obtained with a fine mesh. Finally, a real pipe from the Spanish gas network is solved and the results compared with on-line measurements.

Effect of cladding on thermal behavior of nuclear fuel element with non-uniform heat generation

In nuclear power reactors if the maximum temperature (2000ᵒC) of the fuel element like Uranium-dioxide increments and reaches 2865ᵒC then the fuel element melts down leading to heavy loss and damage. To maintain this maximum temperature well below the allowable limit the heat in the fuel element should be conducted and dissipated to the surrounding medium. In this article, the numerical prediction and comparison of thermal performance characteristics of a nuclear fuel element with and without cladding (bare fuel element) having non-uniform energy generation in axial direction is studied in detail. Accordingly, the two-dimensional heat conduction equation having a cosine variation of heat generation term in the axial direction, along with appropriate boundary conditions is solved using second order accurate finite difference schemes. The results are presented for a wide range of parameters like Aspect Ratio Ar, Biot number, Bi, Conductivity ratio, Cr, Thickness ratio, Th, and total heat generation parameter, Qt in the form of transverse temperature profiles and axial temperature profiles. Effect of Cr and Th due to presence of cladding is thoroughly studied. Different range of parameters for which the temperature of the fuel element is within the maximum temperature range is obtained. Critical values of these parameters for which the temperature increments above maximum temperature is also investigated. From the detailed discussion of the results it is concluded that the maximum temperature attained in the fuel element without cladding (bare fuel element) is very much higher than that attained in fuel element with cladding. Also it is observed that for fixed values of Ar, Cr and Th, there exists a critical value of Qt and Bi beyond and below which maximum temperature in the fuel element exceeds its allowable limit.

High order finite-volume WENO scheme for five-equation model of compressible two-fluid flow

High order finite-volume weighted essentially non-oscillatory (WENO) scheme is applied for solving interface-capturing five-equation model of compressible two-fluid flows in one and two-space dimensions. The model is non-conservative and the governing equations consist of three equations, namely a continuity equation, a momentum equation and an energy balance equation for the fluid mixture and the remaining two are mass and energy equations for one of the two fluids. In the last equation, the non-conservative differential source term appears which is responsible for the energy exchange between fluids. The energy exchange is only due to mechanical work. The presence of non-conservative differential source terms in the two-fluid flow model introduce difficulties in developing high order accurate numerical schemes. The proposed numerical scheme is capable to preserve non-oscillatory property near strong discontinuities and gives high order accuracy in smooth regions. Different one and two-dimensional test problems are considered to analyze efficiency and accuracy of the proposed numerical algorithms. For validation, the solutions of proposed numerical scheme are compared with the results of already available high order kinetic flux-vector splitting scheme and discontinuous Galerkin scheme.

Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine

In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication-avoiding and cache-blocking, and are furthermore very well-suited for vectorization, and so they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing levels of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.

An internal penalty discontinuous Galerkin method for simulating a thermoelectric cooler

Compact modelling of a thermoelectric cooler in the computational fluid dynamics approach is essential for reliable product design in the telecommunication industry, particularly for the package design of the optical laser or the wavelength selective switch. When simulating these fluid thermal structure interaction problems numerically, stability and high order of accuracy are required to capture all of the necessary physics. The principle benefit of using the discontinuous Galerkin (DG) method is that it can produce a high order accurate scheme which can achieve an equivalent error compared to the lower order scheme with orders of magnitude lower computational effort. Based on the recent development of a framework for the computation of fluid thermal structure interaction problems within multi-solid domains using DG methods on unstructured grids([1,2]) ; this paper has proposed a detailed compact thermoelectric cooler (TEC) modelling method based on an existing black box like compact TEC model [4]. Close comparisons validate that both the detailed and the black box like compact model are accurate enough to simulate the conduction only case. When air convection is required to carry out a system-level thermal management optimization, the detailed compact modelling method is more reliable than the black box like compact TEC model. Furthermore, the thermal expansion of an operating TEC has been examined. Compared to the black box like compact TEC model, the simulation results of the detailed compact method have better agreement with electronic speckle pattern interferometry data.

An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation

This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property.

A positivity-preserving high order discontinuous Galerkin scheme for convection–diffusion equations

For constructing high order accurate positivity-preserving schemes for convection–diffusion equations, we construct a simple positivity-preserving diffusion flux. Discontinuous Galerkin (DG) schemes with such a positivity-preserving diffusion flux are nonlinear schemes, which can be regarded as a reduction of the high order positivity-preserving DG schemes for compressible Navier–Stokes equations in [1] to scalar diffusion operators. In this paper we focus on the local DG method to discuss how to apply such a flux. A limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity-preserving in the sense that DG schemes with this flux satisfies a weak positivity property. Together with a positivity-preserving limiter, high order DG schemes with strong stability preserving time discretizations can be rendered positivity-preserving without losing conservation or high order accuracy for convection–diffusion problems with periodic boundary conditions or a special class of Dirichlet or Neumann boundary conditions. Numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme.

A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.

Numerical heat transfer analysis of Maisotsenko Humid Air Bottoming Cycle – A study towards the optimization of the air-water mixture at bottoming turbine inlet

Maisotsenko Humid Air Bottoming Cycle (MHABC) is a recently proposed waste heat recovery unit for gas turbine topping cycle. An air saturator is used for the waste heat recovery and air humidification purposes in the bottoming cycle. The humid air is used in bottoming cycle turbine. In this work, detailed heat transfer analysis of the MHABC is proposed along with the evaluation of degree of humidification and energy recovery. The air saturator is numerically simulated after the development of governing ordinary differential equations for the temperature and humidity distribution of the working air. The governing differential equations of air saturator are discretized using first order accurate finite difference scheme and simulated using MATLAB. The results for the air saturator are initially validated using inlet ambient pressure conditions at standard atmospheric conditions. The upper section of the bottoming cycle is modeled using analytical NTU method. The overall performance of the plant with MHABC is also compared with results reported in literature. Temperature and humidity levels for working air is presented. These results show that the proposed MHABC configuration gives maximum work output (58 MW) and efficiency (∼34%) at the air saturator exit relative humidity of 70%. Corresponding to the maximum system output conditions, the dew point effectiveness of the air saturator is noted at 0.8 with 32% relative humidity at the bottoming turbine inlet. While the air split ratio of 50% between lower air saturator and heat exchanger gives more than 8 MW of waste heat recovery.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Scalar Convection Diffusion Equations

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak...

HEAT TRANSFER IN VISCOPLASTIC BOUNDARY-LAYER FLOW FROM A VERTICAL PERMEABLE CONE WITH MOMENTUM AND THERMAL WALL SLIP: NUMERICAL STUDY

A mathematical model is presented for the laminar free convection boundary layer flow of Casson viscoplastic non-Newtonian fluid external to a vertical penetrable circular cone in the presence of thermal and hydrodynamic slip conditions. The cone surface is maintained at non-uniform surface temperature. The boundary layer conservation equations, which are parabolic in nature, are transformed into non-dimensional form via appropriate similarity variables, and the emerging boundary value problem is solved computationally with the second order accurate implicit Keller-box finite-difference scheme. The influence of velocity (momentum) slip, thermal slip and Casson non-Newtonian parameter on velocity, temperature, skin friction and Nusselt number are illustrated graphically. Validation of solutions with earlier published work is included. The computations show that the flow near the cone surface is strongly decelerated with increasing momentum slip whereas the temperature and thermal boundary layer thickness are increased. Increasing Casson parameter generally decelerates the flow and also decreases temperatures. Both velocity and thermal boundary layer thickness are reduced with greater Prandtl number. The study is relevant to petro-chemical engineering (polymer) processing systems.

DEVELOPMENT OF A HIGH ORDER DISCRETIZATION SCHEME FOR SOLVING FULLY NONLINEAR MAGNETOHYDRODYNAMIC EQUATIONS

We have developed fully fourth order accurate compact finite difference discretization scheme for the Navier-Stokes equations coupled with Maxwell's equations. The implementation is done in cylindrical polar geometry. Due to the full-MHD modeling of physical flow, the modeled equations are fully nonlinear coupled hydrodynamic equations which are again coupled with Maxwells equations. In our computations, we have accounted for the induced magnetic field in the flow of an electrically conducting fluid in an external magnetic field. The code is tested against available experimental and theoretical data where applicable. It is observed that a smaller grid of 64×64 is sufficient for weakly nonlinear problems and higher grids up to 512×512 are needed as the degree of nonlinearities grow in the modeled equation. In the absence of magnetic field, a discontinuity of total drag coefficient and separation length is noted for Re=73 which is in agreement with literature. When the magnetic Reynolds number Rm 1, it decreases nonlinearly, at a much faster rate. Thermal boundary layer thickness decreases as the strength of magnetic field increases and it forces the thermal convection to take place in a laminar structure as observed from thermal contour lines. Finally, using divided differences, we establish that the accuracy of the proposed numerical scheme is in fact fourth order.

Low dissipative entropy stable schemes using third order WENO and TVD reconstructions

A low dissipative framework is given to construct high order entropy stable flux by addition of suitable numerical diffusion operator into entropy conservative flux. The framework is robust in the sense that it allows the use of high order reconstructions which satisfy the sign property only across the discontinuities. The third order weighted essentially non-oscillatory (WENO) interpolations and high order total variation diminishing (TVD) reconstructions are shown to satisfy the sign property across discontinuities. Third order accurate entropy stable schemes are constructed by using third order WENO and high order TVD reconstructions procedures in the diffusion operator. These schemes are efficient and less diffusive since the diffusion is actuated only in the sign stability region of the used reconstruction which includes discontinuities. Numerical results with constructed schemes for various test problems are given which show the third order accuracy and less dissipative nature of the schemes.

Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

In this paper we propose a third order accurate finite volume scheme based on a posteriori limiting of polynomial reconstructions within an Adaptive-Mesh-Refinement (AMR) simulation code for hydrodynamics equations in 2D. The a posteriori limiting is based on the detection of problematic cells on a so-called candidate solution computed at each stage of a third order Runge–Kutta scheme. Such detection may include different properties, derived from physics, such as positivity, from numerics, such as a non-oscillatory behavior, or from computer requirements such as the absence of NaN's. Troubled cell values are discarded and re-computed starting again from the previous time-step using a more dissipative scheme but only locally, close to these cells. By locally decrementing the degree of the polynomial reconstructions from 2 to 0 we switch from a third-order to a first-order accurate but more stable scheme. The entropy indicator sensor is used to refine/coarsen the mesh. This sensor is also employed in an a posteriori manner because if some refinement is needed at the end of a time step, then the current time-step is recomputed with the refined mesh, but only locally, close to the new cells. We show on a large set of numerical tests that this a posteriori limiting procedure coupled with the entropy-based AMR technology can maintain not only optimal accuracy on smooth flows but also stability on discontinuous profiles such as shock waves, contacts, interfaces, etc. Moreover numerical evidences show that this approach is at least comparable in terms of accuracy and cost to a more classical CWENO approach within the same AMR context.

A High Order Well-Balanced Finite Volume WENO Scheme for a Blood Flow Model in Arteries

AbstractThe numerical simulations for the blood flow in arteries by high order accurate schemes have a wide range of applications in medical engineering. The blood flow model admits the steady state solutions, in which the flux gradient is non-zero and is exactly balanced by the source term. In this paper, we present a high order finite volume weighted essentially non-oscillatory (WENO) scheme, which preserves the steady state solutions and maintains genuine high order accuracy for general solutions. The well-balanced property is obtained by a novel source term reformulation and discretisation, combined with well-balanced numerical fluxes. Extensive numerical experiments are carried out to verify well-balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part II, higher order FVTD schemes

The Finite Difference Time Domain (FDTD) scheme has served the computational electrodynamics community very well and part of its success stems from its ability to satisfy the constraints in Maxwell's equations. Even so, in the previous paper of this series we were able to present a second order accurate Godunov scheme for computational electrodynamics (CED) which satisfied all the same constraints and simultaneously retained all the traditional advantages of Godunov schemes. In this paper we extend the Finite Volume Time Domain (FVTD) schemes for CED in material media to better than second order of accuracy.From the FDTD method, we retain a somewhat modified staggering strategy of primal variables which enables a very beneficial constraint-preservation for the electric displacement and magnetic induction vector fields. This is accomplished with constraint-preserving reconstruction methods which are extended in this paper to third and fourth orders of accuracy. The idea of one-dimensional upwinding from Godunov schemes has to be significantly modified to use the multidimensionally upwinded Riemann solvers developed by the first author. In this paper, we show how they can be used within the context of a higher order scheme for CED.We also report on advances in timestepping. We show how Runge–Kutta IMEX schemes can be adapted to CED even in the presence of stiff source terms brought on by large conductivities as well as strong spatial variations in permittivity and permeability. We also formulate very efficient ADER timestepping strategies to endow our method with sub-cell resolving capabilities. As a result, our method can be stiffly-stable and resolve significant sub-cell variation in the material properties within a zone. Moreover, we present ADER schemes that are applicable to all hyperbolic PDEs with stiff source terms and at all orders of accuracy. Our new ADER formulation offers a treatment of stiff source terms that is much more efficient than previous ADER schemes. The computer algebra system scripts for generating ADER time update schemes for any general PDE with stiff source terms are also given in the electronic supplements to this paper.Second, third and fourth order accurate schemes for numerically solving Maxwell's equations in material media are presented in this paper. Several stringent tests are also presented to show that the method works and meets its design goals even when material permittivity and permeability vary by an order of magnitude over just a few zones. Furthermore, since the method is unconditionally stable and sub-cell-resolving in the presence of stiff source terms (i.e. for problems involving giant variations in conductivity over just a few zones), it can accurately handle such problems without any reduction in timestep. We also show that increasing the order of accuracy offers distinct advantages for resolving sub-cell variations in material properties. Most importantly, we show that when the accuracy requirements are stringent the higher order schemes offer the shortest time to solution. This makes a compelling case for the use of higher order, sub-cell resolving schemes in CED.

Numerical modeling of high aspect ratio flexible fibers in inertial flows

A numerical model for the behavior of flexible fibers under inertial flows was developed by coupling discrete element method and finite volume method. The fibers were discretized into several beam segments, and the equations of motion were integrated with a 2nd order accurate explicit scheme. The 3D Navier-Stokes equations were discretized by a 4th order accurate (space and time) unstructured finite volume scheme. Momentum exchange between the fluid and fibers was enforced by including a source term of the fiber hydrodynamic force in the Navier-Stokes equations. The choice of an appropriate model for the hydrodynamic force on a fiber in a fluid flow depending on the Reynolds number is discussed and covers a range of Reynolds number between 10−2 and 102. The current numerical model is validated against different experimental studies, including deflection of fiber in uniform flow, fibers in isotropic turbulent flow, and concentrated fiber suspension in channel flow. The numerical model was able to reproduce the damping/enhancement phenomena of turbulence in a channel flow as a consequence of the micro-structural evolution of the fibers.

High order accurate finite difference schemes based on symmetry preservation

In this paper, we present a mathematical approach that is based on modified equations and the method of equivariant moving frames for construction of high order accurate invariant finite difference schemes that preserve Lie symmetry groups of underlying partial differential equations (PDEs). In the proposed approach, invariant (or symmetry preserving) numerical schemes with a desired (or fixed) order of accuracy are constructed from some non-invariant (base) numerical schemes. Modified forms of PDEs are used to improve the order of accuracy of existing schemes and these modified forms are obtained through addition of defect correction terms to the original forms of PDEs. These defect correction terms of modified PDEs that are noted from truncation error analysis are either completely removed from schemes or their representation is significantly simplified by considering convenient moving frames. This feature of the proposed method can especially be useful to avoid cumbersome numerical representations when high order schemes are developed from low order ones via the method of modified equations. The proposed method is demonstrated via construction of invariant numerical schemes with fixed (and higher) order of accuracy for some common linear and nonlinear problems (including the linear advection–diffusion equation in 1D and 2D, inviscid Burgers' equation, and viscous Burgers' equation) and the performance of these invariant numerical schemes is further evaluated. Our results indicate that such invariant numerical schemes obtained from existing base numerical schemes have the potential to significantly improve the quality of results not only in terms of desired higher order accuracy but also in the context of preservation of appropriate symmetry properties of underlying PDEs.